Project Description
The purpose of this project, was to earn and progress our knowledge on the subject of geometry. In this project, we explored similarity and dilation, through modeling and exploration; we also applied algebra and functions to geometry. Over the course of the project we learned, specifically, about Similarity and Congruence, Proportional Reasoning and the Algebra of Proportions, Polygons and Angles, Logical Reasoning and Proof, Experiments and Data Analysis, and Mathematical Modeling. We not only made sense of the problems and persevered in solving them, but also constructed viable arguments, reasoned abstractly, modeled with mathematic, used tools strategically, attend to precision, looked for and made use of structure, and looked for regularity and patterns among data sets.
We started this project, by summarizing what we had already known about similarity and congruence. Next, we researched and created a poster, on one of the topics that were mentioned above. We got a little preview to the project ahead, and presented our research to the class. After that, we immediately dove into worksheets, and note-taking. We started with rules and equations, then moved onto similar triangles, ladder crossing and dilation. All in the process, we were completing benchmarks one through four, to complete our presentation piece for exhibition.
We started this project, by summarizing what we had already known about similarity and congruence. Next, we researched and created a poster, on one of the topics that were mentioned above. We got a little preview to the project ahead, and presented our research to the class. After that, we immediately dove into worksheets, and note-taking. We started with rules and equations, then moved onto similar triangles, ladder crossing and dilation. All in the process, we were completing benchmarks one through four, to complete our presentation piece for exhibition.
Mathematical Concepts
Congruence and Triangle Congruence
To the left, you can see a worksheet that we completed in class. We're given a series of congruent polygons, and we just have to use equations to find the missing measurements. For number one, we have to solve for t before we can start solving for everything else. Below is the equation I used to find both t and x. Congruence is a lot like Similarity, in the way that a a square can be rectangle. A square can be a rectangle but a rectangle can't be square, just as congruent shapes can be similar, but similar shapes may not always be congruent. The triangle is similar to the smaller triangle, because they are the same shape, but they are congruent, because they have equivalent dimensions. This was one of the main functions I used in benchmark number two, with my partner. When scaling a giraffe down to the size of a small dog is hard, but made a lot easier with this equation, after finding one or two of the giraffes measurements.
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Definition of Similarity
In the picture to the right, you can look at some of my notes on similarity. In basic proportion analysis, two triangles are similar, and we can use proportions to find the missing sides (like we did above). The triangles (on the left) we moved on to were a little more complicated. The approach is the same, you just need to transition from Geometry to Algebra. It's basically finding an equation for your geometrical findings. We can relate similarity to Congruence (as mentioned above), in a way squares and rectangles are related to each other; very similar, but different enough for one to be categorized differently. A giraffe in the wild, and the giraffe my partner and I scaled for benchmark number two are similar. |
Ratios and Proportions (including solving proportions)
To the left, you can see the equation to find proportions. You take your ratios and cross multiply, leaving an algebraic equation that is easy to solve. Proportion relates to dilation, because you are often finding the proportions of dilated shapes. Benchmark number two greatly relied on solving proportions, like represented to the left. |
Proving Similarity: Congruent Angles + Proportional Sides
To the right is a worksheet we completed in class called "Ladder Crossing." The only thing on the worksheet was the two lines crisscrossing, and the measurements of the walls. Dr. Drew told us that finding the height of the intersection is possible, and to start small. In the end, the answer was painfully simple, and we were able to find congruent angles. Congruent angles are a lot like congruent polygons, because they both look pretty intimidating, but can easily be solved by simple steps. My partner and I didn't really use this technique when scaling our giraffe, because giraffes aren't straight edges and sharp corners, they are curved and muscular. |
Dilation, Including Scale Factors and Center of Dilation
Dilation does both ways. Its just shrinking, or make a shape larger. A scale factor is pictured to the left. It shows the ranges of optimal dilation from the center. The center of dilation is often a corner, or the center of the object your making bigger or smaller. Dilation relates to Similarity and Congruence, because the different sizes of shapes that are similar and congruent, have been dilated to that size. My partner and I used dilation to scale down our giraffe to the size of a small dog. |
Dilation: Affect on Distance and Area
To the right is yet another worksheet. When dilation occurs, the size isn't the only thing that changes. More area is taken up, and that needs to accounted for. In the picture, you can see the steady inclination of space taken up by the growing bear. This relates to Dilation, because this is a side effect of dilation. My partner and I didn't use this technique when scaling down the giraffe, but we did use dilation. |
Exhibition
Benchmark #1
Benchmark #1 was an Edmodo-based assignment, and we had to submit answers to the following questions; Who else is on your team? What item/object are you going to scale? How are you going to determine the scale factor? How will your scale model be constructed and exhibited? I worked with my fellow classmate Sammy. We agreed to scale a giraffe down to the size of a small dog. We scaled the actual size of a giraffe down 10x. To exhibit our small giraffe, we put white push pins into a black poster board in the shape of our small giraffe; then wrapped string between the pins, so we have a giraffe outline.
Benchmark #2
Benchmark #2 was not an Edmodo-based assignment. We had to complete our calculations and scaling on paper, then turn them into Dr. Drew's inbox. Below is a picture of our calculations.
Benchmark #1 was an Edmodo-based assignment, and we had to submit answers to the following questions; Who else is on your team? What item/object are you going to scale? How are you going to determine the scale factor? How will your scale model be constructed and exhibited? I worked with my fellow classmate Sammy. We agreed to scale a giraffe down to the size of a small dog. We scaled the actual size of a giraffe down 10x. To exhibit our small giraffe, we put white push pins into a black poster board in the shape of our small giraffe; then wrapped string between the pins, so we have a giraffe outline.
Benchmark #2
Benchmark #2 was not an Edmodo-based assignment. We had to complete our calculations and scaling on paper, then turn them into Dr. Drew's inbox. Below is a picture of our calculations.
Benchmark #3
Benchmark #3 was both Edmodo-based and non-Edmodo-based assignment. On Edmodo, we competed self and partner evaluations. We gave ourself a grade for the past benchmarks, and had to present supporting evidence that showed we were worthy of that grade. The non-Edmodo-based assignment, was just bringing in our exhibition pieces. In our case, the giraffe poster board (below).
Benchmark #3 was both Edmodo-based and non-Edmodo-based assignment. On Edmodo, we competed self and partner evaluations. We gave ourself a grade for the past benchmarks, and had to present supporting evidence that showed we were worthy of that grade. The non-Edmodo-based assignment, was just bringing in our exhibition pieces. In our case, the giraffe poster board (below).
Reflection
This project was very complex, and relied on many moving pieces to flow and reach completion cohesively. As we were reaching due dates for benchmarks, we were learning and completing worksheets on the topics needed for the benchmarks; I stayed surprisingly very organized throughout the entire project. Organization is on of the Habits of a Mathematician that I could improve most on, and this project has definitely made me better at it. I kept every worksheet, hand out, assignment description, and kept them chronologically organized in my binder. That was an area of success. A challenge in this project was applying the worksheet skills to the actual project work. We would do a worksheet in class on a new concept, and then need to apply it that night to submit a benchmark. I had a hard time remembering the skills I needed, and although I had the work with me in my backpack, I didn't think to look it over. Because of this, I had to redo and resubmit some assignments for a better grade. To avoid this in the future, I can study worksheets, find practice problems, perform extensions to better understand, and have my work ready to reference when completing other assignments.